arXiv subjects

Simon Meierhans

Publications and source records attributed to Simon Meierhans.

14 recordsLinked to original sources

arXiv2026

An Approximation Algorithm for Graph Label Selection

In the graph label selection problem, one is given an $n$-vertex graph and a budget $k$, and seeks to select $k$ vertices whose labels enable accurate prediction of the labels on the remaining vertices. This problem formalizes distilling a small representative set from the whole graph. We present the first $\tilde{O}(\log^{1.5} n)$-approximation algorithm for graph label selection under the standard budget constraint. Prior work either relies on resource augmentation, allowing substantially more than $k$ labeled vertices, or consists primarily of heuristics without provable guarantees. Finally, we demonstrate that practical heuristic variants of our algorithm scale to significantly larger graphs than previous methods, while essentially retaining their quality.

cs.DS↗

arXiv2025

Algorithmic Thinking Theory

Large language models (LLMs) have proven to be highly effective for solving complex reasoning tasks. Surprisingly, their capabilities can often be improved by iterating on previously generated solutions. In this context, a reasoning plan for generating and combining a set of solutions can be thought of as an algorithm for reasoning using a probabilistic oracle. We introduce a theoretical framework for analyzing such reasoning algorithms. This framework formalizes the principles underlying popular techniques for iterative improvement and answer aggregation, providing a foundation for designing a new generation of more powerful reasoning methods. Unlike approaches for understanding models that rely on architectural specifics, our model is grounded in experimental evidence. As a result, it offers a general perspective that may extend to a wide range of current and future reasoning oracles.

cs.AI↗

arXiv2025

Dynamic Connectivity with Expected Polylogarithmic Worst-Case Update Time

Whether a graph $G=(V,E)$ is connected is arguably its most fundamental property. Naturally, connectivity was the first characteristic studied for dynamic graphs, i.e. graphs that undergo edge insertions and deletions. While connectivity algorithms with polylogarithmic amortized update time have been known since the 90s, achieving worst-case guarantees has proven more elusive. Two recent breakthroughs have made important progress on this question: (1) Kapron, King and Mountjoy [SODA'13; Best Paper] gave a Monte-Carlo algorithm with polylogarithmic worst-case update time, and (2) Nanongkai, Saranurak and Wulff-Nilsen [STOC'17, FOCS'17] obtained a Las-Vegas data structure, however, with subpolynomial worst-case update time. Their algorithm was subsequently de-randomized [FOCS'20]. In this article, we present a new dynamic connectivity algorithm based on the popular core graph framework that maintains a hierarchy interleaving vertex and edge sparsification. Previous dynamic implementations of the core graph framework required subpolynomial update time. In contrast, we show how to implement it for dynamic connectivity with polylogarithmic expected worst-case update time. We further show that the algorithm can be de-randomized efficiently: a deterministic static algorithm for computing a connectivity edge-sparsifier of low congestion in time $T(m) \cdot m$ on an $m$-edge graph yields a deterministic dynamic connectivity algorithm with $\tilde{O}(T(m))$ worst-case update time. Via current state-of-the-art algorithms [STOC'24], we obtain $T(m) = m^{o(1)}$ and recover deterministic subpolynomial worst-case update time.

cs.DS↗

arXiv2025

Expander Pruning with Polylogarithmic Worst-Case Recourse and Update Time

Expander graphs are known to be robust to edge deletions in the following sense: for any online sequence of edge deletions $e_1, e_2, \ldots, e_k$ to an $m$-edge graph $G$ that is initially a $\phi$-expander, the algorithm can grow a set $P \subseteq V$ such that at any time $t$, $G[V \setminus P]$ is an expander of the same quality as the initial graph $G$ up to a constant factor and the set $P$ has volume at most $O(t/\phi)$. However, currently, there is no algorithm to grow $P$ with low worst-case recourse that achieves any non-trivial guarantee. In this work, we present an algorithm that achieves near-optimal guarantees: we give an algorithm that grows $P$ only by $\tilde{O}(1/\phi^2)$ vertices per time step and ensures that $G[V \setminus P]$ remains $\tilde{\Omega}(\phi)$-expander at any time. Even more excitingly, our algorithm is extremely efficient: it can process each update in near-optimal worst-case update time $\tilde{O}(1/\phi^2)$. This affirmatively answers the main open question posed in [SW19] whether such an algorithm exists. By combining our results with recent techniques in [BvdBPG+22], we obtain the first adaptive algorithms to maintain spanners, cut and spectral sparsifiers with $\tilde{O}(n)$ edges and polylogarithmic approximation guarantees, worst-case update time and recourse. More generally, we believe that worst-case pruning is an essential tool for obtaining worst-case guarantees in dynamic graph algorithms and online algorithms.

cs.DS↗

arXiv2024

Parallel and Distributed Expander Decomposition: Simple, Fast, and Near-Optimal

Expander decompositions have become one of the central frameworks in the design of fast algorithms. For an undirected graph $G=(V,E)$, a near-optimal $\phi$-expander decomposition is a partition $V_1, V_2, \ldots, V_k$ of the vertex set $V$ where each subgraph $G[V_i]$ is a $\phi$-expander, and only an $\widetilde{O}(\phi)$-fraction of the edges cross between partition sets. In this article, we give the first near-optimal parallel algorithm to compute $\phi$-expander decompositions in near-linear work $\widetilde{O}(m/\phi^2)$ and near-constant span $\widetilde{O}(1/\phi^4)$. Our algorithm is very simple and likely practical. Our algorithm can also be implemented in the distributed Congest model in $\tilde{O}(1/\phi^4)$ rounds. Our results surpass the theoretical guarantees of the current state-of-the-art parallel algorithms [Chang-Saranurak PODC'19, Chang-Saranurak FOCS'20], while being the first to ensure that only an $\tilde{O}(\phi)$ fraction of edges cross between partition sets. In contrast, previous algorithms [Chang-Saranurak PODC'19, Chang-Saranurak FOCS'20] admit at least an $O(\phi^{1/3})$ fraction of crossing edges, a polynomial loss in quality inherent to their random-walk-based techniques. Our algorithm, instead, leverages flow-based techniques and extends the popular sequential algorithm presented in [Saranurak-Wang SODA'19].

cs.DS↗

arXiv2024

A Simple Dynamic Spanner via APSP

We give a simple algorithm for maintaining a $n^{o(1)}$-approximate spanner $H$ of a graph $G$ with $n$ vertices as $G$ receives edge updates by reduction to the dynamic All-Pairs Shortest Paths (APSP) problem. Given an initially empty graph $G$, our algorithm processes $m$ insertions and $n$ deletions in total time $m^{1 + o(1)}$ and maintains an initially empty spanner $H$ with total recourse $n^{1 + o(1)}$. When the number of insertions is much larger than the number of deletions, this notably yields recourse sub-linear in the total number of updates. Our algorithm only has a single $O(\log n)$ factor overhead in runtime and approximation compared to the underlying APSP data structure. Therefore, future improvements for APSP will directly yield an improved dynamic spanner.

cs.DS↗

arXiv2024

Bootstrapping Dynamic APSP via Sparsification

We give a simple algorithm for the dynamic approximate All-Pairs Shortest Paths (APSP) problem. Given a graph $G = (V, E, l)$ with polynomially bounded edge lengths, our data structure processes $|E|$ edge insertions and deletions in total time $|E|^{1 + o(1)}$ and provides query access to $|E|^{o(1)}$-approximate distances in time $\tilde{O}(1)$ per query. We produce a data structure that mimics Thorup-Zwick distance oracles [TZ'05], but is dynamic and deterministic. Our algorithm selects a small number of pivot vertices. Then, for every other vertex, it reduces distance computation to maintaining distances to a small neighborhood around that vertex and to the nearest pivot. We maintain distances between pivots efficiently by representing them in a smaller graph and recursing. We construct these smaller graphs by (a) reducing vertex count using the dynamic distance-preserving core graphs of Kyng-Meierhans-Probst Gutenberg [KMPG'24] in a black-box manner and (b) reducing edge-count using a dynamic spanner akin to Chen-Kyng-Liu-Meierhans-Probst Gutenberg [CKL+'24]. Our dynamic spanner internally uses an APSP data structure. Choosing a large enough size reduction factor in the first step allows us to simultaneously bootstrap our spanner and a dynamic APSP data structure. Notably, our approach does not need expander graphs, an otherwise ubiquitous tool in derandomization.

cs.DS↗

arXiv2024

Almost-Linear Time Algorithms for Decremental Graphs: Min-Cost Flow and More via Duality

We give the first almost-linear total time algorithm for deciding if a flow of cost at most $F$ still exists in a directed graph, with edge costs and capacities, undergoing decremental updates, i.e., edge deletions, capacity decreases, and cost increases. This implies almost-linear time algorithms for approximating the minimum-cost flow value and $s$-$t$ distance on such decremental graphs. Our framework additionally allows us to maintain decremental strongly connected components in almost-linear time deterministically. These algorithms also improve over the current best known runtimes for statically computing minimum-cost flow, in both the randomized and deterministic settings. We obtain our algorithms by taking the dual perspective, which yields cut-based algorithms. More precisely, our algorithm computes the flow via a sequence of $m^{1+o(1)}$ dynamic min-ratio cut problems, the dual analog of the dynamic min-ratio cycle problem that underlies recent fast algorithms for minimum-cost flow. Our main technical contribution is a new data structure that returns an approximately optimal min-ratio cut in amortized $m^{o(1)}$ time by maintaining a tree-cut sparsifier. This is achieved by devising a new algorithm to maintain the dynamic expander hierarchy of [Goranci-R\"{a}cke-Saranurak-Tan, SODA 2021] that also works in capacitated graphs. All our algorithms are deterministc, though they can be sped up further using randomized techniques while still working against an adaptive adversary.

cs.DS↗

arXiv2023

Almost-Linear Time Algorithms for Incremental Graphs: Cycle Detection, SCCs, $s$-$t$ Shortest Path, and Minimum-Cost Flow

We give the first almost-linear time algorithms for several problems in incremental graphs including cycle detection, strongly connected component maintenance, $s$-$t$ shortest path, maximum flow, and minimum-cost flow. To solve these problems, we give a deterministic data structure that returns a $m^{o(1)}$-approximate minimum-ratio cycle in fully dynamic graphs in amortized $m^{o(1)}$ time per update. Combining this with the interior point method framework of Brand-Liu-Sidford (STOC 2023) gives the first almost-linear time algorithm for deciding the first update in an incremental graph after which the cost of the minimum-cost flow attains value at most some given threshold $F$. By rather direct reductions to minimum-cost flow, we are then able to solve the problems in incremental graphs mentioned above. At a high level, our algorithm dynamizes the $\ell_1$ oblivious routing of Rozhoň-Grunau-Haeupler-Zuzic-Li (STOC 2022), and develops a method to extract an approximate minimum ratio cycle from the structure of the oblivious routing. To maintain the oblivious routing, we use tools from concurrent work of Kyng-Meierhans-Probst Gutenberg which designed vertex sparsifiers for shortest paths, in order to maintain a sparse neighborhood cover in fully dynamic graphs. To find a cycle, we first show that an approximate minimum ratio cycle can be represented as a fundamental cycle on a small set of trees resulting from the oblivious routing. Then, we find a cycle whose quality is comparable to the best tree cycle. This final cycle query step involves vertex and edge sparsification procedures reminiscent of previous works, but crucially requires a more powerful dynamic spanner which can handle far more edge insertions. We build such a spanner via a construction that hearkens back to the classic greedy spanner algorithm.

cs.DS↗

arXiv2023

A Dynamic Shortest Paths Toolbox: Low-Congestion Vertex Sparsifiers and their Applications

We present a general toolbox, based on new vertex sparsifiers, for designing data structures to maintain shortest paths in dynamic graphs. In an $m$-edge graph undergoing edge insertions and deletions, our data structures give the first algorithms for maintaining (a) $m^{o(1)}$-approximate all-pairs shortest paths (APSP) with \emph{worst-case} update time $m^{o(1)}$ and query time $\tilde{O}(1)$, and (b) a tree $T$ that has diameter no larger than a subpolynomial factor times the diameter of the underlying graph, where each update is handled in amortized subpolynomial time. In graphs undergoing only edge deletions, we develop a simpler and more efficient data structure to maintain a $(1+ε)$-approximate single-source shortest paths (SSSP) tree $T$ in a graph undergoing edge deletions in amortized time $m^{o(1)}$ per update. Our data structures are deterministic. The trees we can maintain are not subgraphs of $G$, but embed with small edge congestion into $G$. This is in stark contrast to previous approaches and is useful for algorithms that internally use trees to route flow. To illustrate the power of our new toolbox, we show that our SSSP data structure gives simple deterministic implementations of flow-routing MWU methods in several contexts, where previously only randomized methods had been known. To obtain our toolbox, we give the first algorithm that, given a graph $G$ undergoing edge insertions and deletions and a dynamic terminal set $A$, maintains a vertex sparsifier $H$ that approximately preserves distances between terminals in $A$, consists of at most $|A|m^{o(1)}$ vertices and edges, and can be updated in worst-case time $m^{o(1)}$. Crucially, our vertex sparsifier construction allows us to maintain a low edge-congestion embedding of $H$ into $G$, which is needed for our applications.

cs.DS↗

arXiv2022

Derandomizing Directed Random Walks in Almost-Linear Time

In this article, we present the first deterministic directed Laplacian L systems solver that runs in time almost-linear in the number of non-zero entries of L. Previous reductions imply the first deterministic almost-linear time algorithms for computing various fundamental quantities on directed graphs including stationary distributions, personalized PageRank, hitting times and escape probabilities. We obtain these results by introducing partial symmetrization, a new technique that makes the Laplacian of an Eulerian directed graph ``less directed'' in a useful sense, which may be of independent interest. The usefulness of this technique comes from two key observations: Firstly, the partially symmetrized Laplacian preconditions the original Eulerian Laplacian well in Richardson iteration, enabling us to construct a solver for the original matrix from a solver for the partially symmetrized one. Secondly, the undirected structure in the partially symmetrized Laplacian makes it possible to sparsify the matrix very crudely, i.e. with large spectral error, and still show that Richardson iterations convergence when using the sparsified matrix as a preconditioner. This allows us to develop deterministic sparsification tools for the partially symmetrized Laplacian. Together with previous reductions from directed Laplacians to Eulerian Laplacians, our technique results in the first deterministic almost-linear time algorithm for solving linear equations in directed Laplacians. To emphasize the generality of our new technique, we show that two prominent existing (randomized) frameworks for solving linear equations in Eulerian Laplacians can be derandomized in this way: the squaring-based framework of Cohen, Kelner, Peebles, Peng, Rao, Sidford and Vladu (STOC 2017) and the sparsified Cholesky-based framework of Peng and Song (STOC 2022).

cs.DS↗

arXiv2021

Incremental SSSP for Sparse Digraphs Beyond the Hopset Barrier

Given a directed, weighted graph $G=(V,E)$ undergoing edge insertions, the incremental single-source shortest paths (SSSP) problem asks for the maintenance of approximate distances from a dedicated source $s$ while optimizing the total time required to process the insertion sequence of $m$ edges. Recently, Gutenberg, Williams and Wein [STOC'20] introduced a deterministic $\tilde{O}(n^2)$ algorithm for this problem, achieving near linear time for very dense graphs. For sparse graphs, Chechik and Zhang [SODA'21] recently presented a deterministic $\tilde{O}(m^{5/3})$ algorithm, and an adaptive randomized algorithm with run-time $\tilde{O}(m\sqrt{n} + m^{7/5})$. This algorithm is remarkable for two reasons: 1) in very spare graphs it reaches the directed hopset barrier of $\tildeΩ(n^{3/2})$ that applied to all previous approaches for partially-dynamic SSSP [STOC'14, SODA'20, FOCS'20] \emph{and} 2) it does not resort to a directed hopset technique itself. In this article we introduce \emph{propagation synchronization}, a new technique for controlling the error build-up on paths throughout batches of insertions. This leads us to a significant improvement of the approach in [SODA'21] yielding a \emph{deterministic} $\tilde{O}(m^{3/2})$ algorithm for the problem. By a very careful combination of our new technique with the sampling approach from [SODA'21], we further obtain an adaptive randomized algorithm with total update time $\tilde{O}(m^{4/3})$. This is the first partially-dynamic SSSP algorithm in sparse graphs to bypass the notorious directed hopset barrier which is often seen as the fundamental challenge towards achieving truly near-linear time algorithms.

cs.DS↗

arXiv2020

Transformations of High-Level Synthesis Codes for High-Performance Computing

Spatial computing architectures promise a major stride in performance and energy efficiency over the traditional load/store devices currently employed in large scale computing systems. The adoption of high-level synthesis (HLS) from languages such as C++ and OpenCL has greatly increased programmer productivity when designing for such platforms. While this has enabled a wider audience to target spatial computing architectures, the optimization principles known from traditional software design are no longer sufficient to implement high-performance codes, due to fundamentally distinct aspects of hardware design, such as programming for deep pipelines, distributed memory resources, and scalable routing. To alleviate this, we present a collection of optimizing transformations for HLS, targeting scalable and efficient architectures for high-performance computing (HPC) applications. We systematically identify classes of transformations (pipelining, scalability, and memory), the characteristics of their effect on the HLS code and the resulting hardware (e.g., increasing data reuse or resource consumption), and the objectives that each transformation can target (e.g., resolve interface contention, or increase parallelism). We show how these can be used to efficiently exploit pipelining, on-chip distributed fast memory, and on-chip dataflow, allowing for massively parallel architectures. To quantify the effect of various transformations, we cover the optimization process of a sample set of HPC kernels, provided as open source reference codes. We aim to establish a common toolbox to guide both performance engineers and compiler engineers in tapping into the performance potential offered by spatial computing architectures using HLS.

cs.DC↗

arXiv2019

Content Adaptive Optimization for Neural Image Compression

The field of neural image compression has witnessed exciting progress as recently proposed architectures already surpass the established transform coding based approaches. While, so far, research has mainly focused on architecture and model improvements, in this work we explore content adaptive optimization. To this end, we introduce an iterative procedure which adapts the latent representation to the specific content we wish to compress while keeping the parameters of the network and the predictive model fixed. Our experiments show that this allows for an overall increase in rate-distortion performance, independently of the specific architecture used. Furthermore, we also evaluate this strategy in the context of adapting a pretrained network to other content that is different in visual appearance or resolution. Here, our experiments show that our adaptation strategy can largely close the gap as compared to models specifically trained for the given content while having the benefit that no additional data in the form of model parameter updates has to be transmitted.

cs.CV↗